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# Solution

The registration number of the motorcycle is a four-digit number. Let the number be wxyz.

The first three digits are in an arithmetic sequence, and the last three digits are in a geometric progression. It means w, x, and y are in AP, and x, y, and z are in GP.

Let's go step by step:

1. Now, w, x, and y are in AP. So, here the first term of AP is w, the second term is x, and the third term is y. The common difference, d = 2. We know that the nth term of AP is [a + (n - 1)d], where a is the first term and d is the common difference. So, in this case, x = [w + (2 - 1)*2] and y = [w + (3 - 1)*2]. So, x = w + 2 and y = w + 4.

2. So, our digits are w, (w + 2), (w + 4), z.

3. The last digit is four times the first digit plus one. So, z = (4w + 1).

4. So, now our sequence of digits is w, (w + 2), (w + 4), (4w + 1).

5. The last three digits are in GP; so, (w + 2), (w + 4), (4w + 1) are in GP. So, if the common ratio = r, then (w + 4) / (w + 2) = r = (4w + 1) / (w + 4).

6. So, (w + 4) / (w + 2) = (4w + 1) / (w + 4).

7. So, (w + 4)^{2} = (4w + 1) * (w + 2).

8. On expanding the terms, we get w^{2} + 8w + 16 = 4w^{2} + 8w + w + 2.

9. So, w^{2} + 8w + 16 = 4w^{2} + 9w + 2.

10. So, 3w^{2} + w - 14 = 0.

11. Solving this equation, we get 3w^{2} + 7w - 6w - 14 = 0. That gives w (3w + 7) - 2 (3w + 7) = 0. So, (3w + 7) * (w - 2) = 0.

12. So, (3w + 7) = 0 or (w - 2) = 0. So, w = -7/3 or w = 2.

13. In our case, w is the first digit of the registration number; it cannot be negative. So, w = 2.

14. The four digits are w, (w + 2), (w + 4), (4w + 1). So, the digits are 2, 4, 6, and 9.

15. Therefore, the registration number of the motorcycle is 2469.

**Some useful links for your career:**

- Union Public Service Commission - www.upsc.gov.in
- IIT-Kharagpur - www.iitkgp.ac.in
- Indian Statistical Institute - www.isical.ac.in
- Indian Institute of Technology Madras - www.iitm.ac.in
- Indian Institute of Management, Ahmedabad - www.iimahd.ernet.in
- Indian Institute of Mass Commission - www.iimc.nic.in
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- Indian School of Mines, Dhanbad - www.ismdhanbad.ac.in
- Birla Institute of Technology, Ranchi - www.bitmesra.ac.in
- Central Institute of Fisheries Nautical and Engineering Training - www.cifnet.nic.in
- Indian Institute of Information Technology, Allahabad (Deemed University) - www.iiita.ac.in
- Central Marine Fisheries Research Institute, Kochi - www.cmfri.com
- Tata Institute of Social Sciences, Mumbai - www.tiss.edu